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C. Theorems of Continuity

THEOREM–1 If f & g are two functions that are continuous at x= c then the functions defined by F1(x) = f(x) ± g(x) ; F2(x) = K f(x)  K any real number ; F3(x) = f(x).g(x) are also continuous at x= c.
Further, if g (c) is not zero, then  Theorems of Continuity | Mathematics (Maths) Class 12 - JEE  is also continuous at x = c.

THEOREM–2 If  f(x) is continuous & g(x) is discontinuous at x = a  then the product function ø(x) = f(x) . g(x) is not necessarily discontinuous at x = a .

Theorems of Continuity | Mathematics (Maths) Class 12 - JEE

THEOREM–3 If f(x) and g(x)  both  are  discontinuous  at  x = a  then  the  product  function ø(x) = f(x) . g(x) is not necessarily discontinuous at  x = a .

Theorems of Continuity | Mathematics (Maths) Class 12 - JEE

Theorems-4 : Intermediate Value Theorem 

If f is continuous on the closed interval [a, b] and k is any number between f(a) and f(b), then there is at least one number c in [a, b] such that f(c) = k.

Note that the Intermediate Value Theorem tells that at least one c exists, but it does not give a method for finding c. Such theorems are called existence theorems.

As a simple example of this theorem, consider a person's height. Suppose that a girl is 5 feet tall on her thirteenth birthday and 5 feet 7 inches tall on her fourteenth birthday. Then, for any height h between 5 feet and 7 inches, there must have been a time t when her height was exactly h. This seems reasonable because human growth is continuous and a person's height does not abruptly change from one value to another.

The Intermediate Value Theorem guarantees the existence of at least one number c in the closed interval [a, b]. There may, of course, be more than one number c such that f(c) = k, as shown in Figure 1. A function that is not continuous does not necessarily possess the intermediate value property. For example, the graph of the function shown in Figure 2 jumps over the horizontal line given by y = k and for this function there is no value of c in [a, b] such that f(c) = k.

The Intermediate Value Theorem often can be used to locate the zeroes of a function that is continuous on a closed interval. Specifically, if f is continuous on [a, b] and f(a) and f(b) differ in sign, then the intermediate Value Theorem guarantees the existence of at least one zero of f in the closed interval [a, b].

Theorems of Continuity | Mathematics (Maths) Class 12 - JEE   Theorems of Continuity | Mathematics (Maths) Class 12 - JEE   Theorems of Continuity | Mathematics (Maths) Class 12 - JEE

 

Ex.10 Use the Intermediate Value Theorem to show that the polynomial function f(x) = x3 + 2x - 1 has a zero in the interval [0, 1]

Sol. Note that f is continuous on the closed interval [0, 1]. Because

f(0) = 03 + 2(0) - 1 = -1         and                f(1) = 13 + 2(1) - 1 = 2

it follows that f(0) < 0 and f(1) > 0. You can therefore apply the Intermediate Value Theorem to conclude that there must be some c in [0, 1] such that f(c) = 0, as shown in Figure 3.

Ex.11 State intermediate value theorem and use it to prove that the equation Theorems of Continuity | Mathematics (Maths) Class 12 - JEE  has at least one real root.

Sol. Let f (x) = Theorems of Continuity | Mathematics (Maths) Class 12 - JEE   first, f (x) is continuous on [5, 6]

Theorems of Continuity | Mathematics (Maths) Class 12 - JEE

Theorems of Continuity | Mathematics (Maths) Class 12 - JEE

Theorems of Continuity | Mathematics (Maths) Class 12 - JEE

Hence by intermediate value theorem É at least one value of c ∈ (5, 6) for which f (c) = 0

Theorems of Continuity | Mathematics (Maths) Class 12 - JEE

 c is root of the equation  Theorems of Continuity | Mathematics (Maths) Class 12 - JEE

 

Ex.12 If f(x) be a continuous function in Theorems of Continuity | Mathematics (Maths) Class 12 - JEE then prove that there exists point Theorems of Continuity | Mathematics (Maths) Class 12 - JEE

Sol. 

Let g(x) = f(x) – f(x + π) ....(i)

at x = π; g(π) = f(π) – f(2π) ....(ii)

at x = 0, g(0) = f(0) – f(π) ...(iii)

adding (ii) and (iii), g(0) + g(π) = f(0) – f(2π)

⇒ g(0) + g(π) = 0 [Given f(0) = f(2π) ⇒ g(0) = –g(π)

⇒ g(0) and g(π) are opposite in sign.

⇒ There exists a point c between 0 and p such g(c) = 0 as shown in graph;

From (i) putting x = c g(c) = f(c) – f(c +π) = 0 Hence, f(c) = f(c + π)

The document Theorems of Continuity | Mathematics (Maths) Class 12 - JEE is a part of the JEE Course Mathematics (Maths) Class 12.
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FAQs on Theorems of Continuity - Mathematics (Maths) Class 12 - JEE

1. What is the definition of continuity in mathematics?
Ans. Continuity in mathematics refers to the property of a function where small changes in the input result in small changes in the output. Formally, a function f(x) is continuous at a point x=c if the limit of f(x) as x approaches c exists and is equal to f(c).
2. How do you prove continuity using the epsilon-delta definition?
Ans. To prove continuity using the epsilon-delta definition, you need to show that for any given epsilon (ε) greater than zero, there exists a delta (δ) greater than zero such that if the absolute value of (x-c) is less than delta, then the absolute value of (f(x)-f(c)) is less than epsilon. This ensures that small changes in the input result in small changes in the output.
3. What is the Intermediate Value Theorem?
Ans. The Intermediate Value Theorem states that if a function f(x) is continuous on a closed interval [a, b] and takes on two different values, say f(a) and f(b), then it must take on every value between f(a) and f(b) at least once on that interval. In other words, the function must cross every intermediate value between its endpoints.
4. How does the Bolzano-Weierstrass Theorem relate to continuity?
Ans. The Bolzano-Weierstrass Theorem states that any bounded sequence of real numbers has a convergent subsequence. This theorem is related to continuity because it helps establish the existence of a limit for a continuous function. If a sequence is bounded and its terms are in the range of a continuous function, then the Bolzano-Weierstrass Theorem guarantees the existence of a limit for that function.
5. Can a function be continuous at some points and discontinuous at others?
Ans. Yes, it is possible for a function to be continuous at certain points and discontinuous at others. This occurs when the function satisfies the conditions for continuity at some points, but fails to satisfy them at others. A function can exhibit different types of discontinuities, such as jump discontinuities, removable discontinuities, or oscillating discontinuities, while still being continuous at other points.
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